I'm trying to export a character to UE4, but I can't seem to export the shape key animation and bone constraints, how could I get around this?
The usual workflow to export animation with constraints is to perform the animation in Blender, then select all deforming bones and bake the animation (Pose mode, pose menu, animation, bake action), visual transform, delete constraints.
In this way you'll export an armature with no constraints that can be accepted by all others softwares.
Be sure to perform the bake on a copy of your file, because it's a destructive edit.